The Journal of Symbolic Logic
Volume 73, Number 2, June 2008
RANDOMNESS, LOWNESS AND DEGREES
GEORGE BARMPALIAS, ANDREW E. M. LEWIS, AND MARIYA SOSKOVA
Abstract. We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if
oracle A can identify some patterns on some real !, oracle B can also find patterns on !. In other words, B
is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e.,
restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among
other results we show that whenever α is not GL2 the LR degree of α bounds 2ℵ0 degrees (so that, in
particular, there exist LR degrees with uncountably many predecessors) and we give sample results which
demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about
the c.e. LR degrees.
§1. Introduction. One of the goals of this work is to present a uniform approach
to studying the LR degrees, both globally and locally. So far the known results about
this degree structure have mostly been scattered and in papers dealing with a wider
range of themes in algorithmic randomness (see for example [15]). An exception
is Simpson’s recent paper [20] which deals with themes like almost everywhere
domination which are very closely related to the LR degrees. Also, a number of
results in this area have been proved via a mix of frameworks like martingales, prefixfree complexity and Martin-Löf tests, with more than one framework sometimes
appearing in the same proof (see [15, 14]). In contrast, we present proofs of new
and old results using only the Martin-Löf approach, i.e., Σ01 classes and (in the
relativised case) c.e. operators. We work in the Cantor space 2# with the usual
topology generated by the basic open intervals
[$] = {% | % ∈ 2# ∧ $ ⊆ %}
(where $ is a finite binary string and $ ⊆ % denotes that $ is a prefix of %) and the
Lebesgue measure generated by &([$]) = 2−|$| . We systematically confuse sets of
finite strings U with the class of reals which extend some string in U . Thus
• we write &(U ) for the measure of the corresponding class of reals
• all subset relations U ⊂ V where U, V are sets of strings actually refer to the
corresponding classes of reals
Received December 30, 2006.
Barmpalias was supported by EPSRC Research Grant No. EP/C001389/1 and would also like to
thank Steve Simpson for a number of stimulating discussions, as well as Doug Cenzer for his hospitality
during a visit to the University of Florida, September-November 2006. Lewis was supported by MarieCurie Fellowship No. MEIF-CT-2005-023657. Soskova was supported by the Marie Curie Early Training
grant MATHLOGAPS (MEST-CT-2004-504029). All authors were partially supported by the NSFC
Grand International Joint Project, No. 60310213, New Directions in the Theory and Applications of
Models of Computation.
c 2008, Association for Symbolic Logic
!
0022-4812/08/7302-0012/$2.90
559
560
G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
• Boolean operations on sets of strings actually refer to the same operations on
the corresponding classes of reals.
In Section 2 we review the basic definition of an oracle Martin-Löf test and make
the simple observation that there exists a universal test with certain nice properties
that will later be useful. It is worth noting that not all universal (even unrelativised)
Martin-Löf tests have the same properties and for some arguments it is convenient
to assume that we hold a special test. This is not new—see for example [11] where
a property of the test derived by its construction (and not the general definition) is
used to show that random sets are effectively immune. We also briefly survey a few
useful known results on Martin-Löf randomness.
In Section 3 we give the definition of ≤LR and the induced degree structure
and mention some known properties. We give a slightly different proof of KjosHansen’s characterization of ≤LR in terms of relativised Σ01 classes of small measure
and establish a connection between any two universal oracle tests. We also present
Frank Stephan’s proof (delivered after a query of the first author) that non-trivial
≤LR -upper cones have measure 0. In Section 4 we show that there is a continuum of
reals which are LR-reducible to the halting problem and then extend this argument
to show that the same is true of any α which is not GL2 .
In Section 5 we study the structure of the computably enumerable LR degrees.
The main goal here is to show how techniques from the theory of the c.e. Turing
degrees can be transferred to the c.e. LR degrees. We deal with two fundamental
techniques: Sacks coding and Sacks restraints. First we show that if A has intermediate c.e. Turing degree then the lower cone of c.e. LR degrees below it properly
extends the corresponding cone of c.e. Turing degrees. The second example demonstrates the use of Sacks restraints in the LR context and is a splitting theorem for
the LR degrees: every c.e. set can be split into two c.e. sets in a non-trivial way with
respect to LR equivalence. Both of these results concern the relationship between
the LR and the Turing degrees. Section 6 is a note on enumerations of random sets.
We show that the relativised arithmetical and Ershov hierarchies are still proper
when restricted to the class of random sets. Finally in Section 7 we present an adaptation of Nies’ original argument that low for random sets are ∆02 which is phrased
entirely in the framework of Martin-Löf tests and provides some information about
universal tests of a certain type.
§2. Oracle Martin-Löf tests. An oracle Martin-Löf test (Ue ) is a uniform sequence of oracle machines which output finite binary strings such that if Ue% is
the range of the e-th machine with oracle % ∈ 2# then for all % ∈ 2# , e ∈ N,
%
&(Ue% ) < 2−(e+1) and Ue% ⊇ Ue+1
. A real α is called %-random if for every oracle
Martin-Löf test (Ue ) we have α ∈
/ ∩e Ue% . A universal oracle Martin-Löf test is
an oracle Martin-Löf test (Ue ) such that for every α, % ∈ 2# , α is %-random iff
α (∈ ∩e Ue% . Given any oracle Martin-Löf test (Ue ), each Ue can be thought of as a
c.e. set of axioms )', $*. If % ∈ 2# then
Ue% = {$ | ∃'(' ⊂ % ∧ )', $* ∈ Ue )}
and for ( ∈ 2<# we define
Ue( = {$ | ∃'(' ⊆ ( ∧ )', $* ∈ Ue )}.
RANDOMNESS, LOWNESS AND DEGREES
561
There is an analogy between oracle Martin-Löf tests as defined above and Lachlan
functionals i.e., Turing functionals viewed as c.e. sets of axioms. This analogy will
be exploited in a number of constructions below, especially in the constructions of
c.e. LR degrees. The following lemma is easily proved and provides a universal
oracle Martin-Löf test with properties which will later be useful.
Lemma 2.1. There is an oracle Martin-Löf test (Ue ) such that
• For every oracle Martin-Löf test (Ve ), uniformly on its c.e. index we can compute
%
k ∈ N such that for every real % and all e, Ve+k
⊆ Ue% .
• If )'1 , $1 *, )'2 , $2 * ∈ Ue and '1 ⊆ '2 then $1 |$2 .
• If )', $* ∈ Ue then |'| = |$| and )', $* ∈ Ue [|'|] − Ue [|'| − 1].
Proof. Suppose that Φi is an effective enumeration of all oracle Martin-Löf tests,
so that Φi (e) is the e-th member of the i-th oracle Martin-Löf test. We enumerate
a test such that the above conditions are satisfied. Given e we wish to enumerate
into the e-th member of the test all )', $* which belong to Φi (e + i + 1). This would
give an oracle Martin-Löf test (Ee ) which satisfies the first condition. To satisfy the
second and third conditions as well we do the same but with a modification. If at
some stage s we would enumerate )', $* into Ee according to the construction just
described, now we split ' into a prefix-free finite set B of strings of length s (i.e., B
is the set of strings of length s which extend '), $ into a prefix-free finite set C of
strings of length s and replace )', $* with the axioms )' # , $ # * for all ' # ∈ B, $ # ∈ C .
We enumerate into Ue each string )' # , $ # * with ' # ∈ B, $ # ∈ C such that for every
)(, )* currently in Ue , ( (⊆ ' # or ) (⊆ $ # . It is easy to verify that the new test (Ue ) is
essentially the same as the simpler (Ee ) in the sense that for every % ∈ 2# and e ∈ N
the class of reals in Ue% is the same as the class of reals determined by Ee% .
Corollary 2.2. Let (Ue ) be the universal oracle Martin-Löf test of Lemma 2.1
and let U be any member of it. There is a computable function which, given any
"
input )', ' # * such that ' ⊆ ' # , outputs the finite (clopen) set U ' − U ' .
Proof. This follows from the properties of (Ue ) as described in Lemma 2.1. If U, V are two prefix-free sets of strings it is not hard to show that UV is prefix
free and &(UV ) = &(U ) · &(V ), where
UV = {$' | $ ∈ U ∧ ' ∈ V }.
Hence for any n, &(U n ) = (&(U ))n . However if U is not prefix-free this may
not hold (it is not hard to find a counterexample). For this reason, wherever we
consider such products below we always assume that the underlying sets are prefixfree. It is also worth noting that if U1 , U2 are two different prefix-free sets of strings
representing the same open set of reals, U12 does not necessarily represent the same
set of reals as U22 (e.g., consider the case where U1 = {0} and U2 = {00, 01}).
However this fact will not cause any problems in what follows as we only deal with
a single representation for a given open set.
In the following we survey some fundamental known results which are needed
in order to understand Bjørn Kjos-Hanssen’s characterisation of ≤LR which is
presented in Section 3. Every Π01 class containing a random real must have positive
measure. In order to see this observe that otherwise we would be able to use this
Π01 class in order to define a Martin-Löf test which contains precisely the members
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G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
of that Π01 class, giving a contradiction. This observation clearly relativises to any
oracle B. This implies that if (VeB ) is a universal Martin-Löf test relative to B then
for every e and $ either [$] ⊆ VeB or
&([$] ∩ VeB )
<1
(1)
2−|$|
(note that the equality above denotes a definition of the leftmost term). In order to
see this observe that if [$] (⊆ VeB then the reals which extend $ and do not belong
to VeB are all B-random and form a nonempty Π01 class relative to B. This class
must therefore have positive measure. Finally it is useful to know that if P is a
Π01 class of positive measure, then every random real has a tail in P [11]. Indeed,
consider the complement V of P as a prefix-free c.e. set of strings and a random
%. Then &(V ) < 1 and there exists a computable function f such that (V f(n) ) is a
Martin-Löf test. Thus there exists a least n such that % (∈ V n and a $ ∈ V n−1 such
that $ ⊂ %. This means that the tail of % starting from position |$| belongs to P.
This observation relativises to any oracle.
&$ (VeB ) :=
§3. LR reducibility and degrees. The LR reducibility was introduced in [16].
Definition 1. [16] Let A ≤LR B if every B-random real is A-random. The
induced degree structure is called the LR degrees.
Intuitively this means that if oracle A can identify some patterns on some real !,
oracle B can also find patterns on !. In other words, B is at least as good as A for
this purpose. It is not hard to show (especially in view of Theorem 3.2) that ≤LR
is Σ03 definable and this has been noticed by a number of authors. Being Σ03 means
that it has some things in common with ≤T (which is also Σ03 ) and this can be seen
more clearly in Section 5 where techniques from the theory of c.e. Turing degrees
are seen to be applicable in the c.e. LR degrees. For more examples of similar Σ03
relations see [20]. We point out (after [16, 20]) that a strict relativization of the
notion of low-for-random [12] gives that A is low-for-random relative to B when
A ⊕ B ≤LR B, which is different than A ≤LR B. In particular, ⊕ does not define
join in the LR degrees and it is an open question as to whether any two degrees
always have a least upper bound in this structure [16, 20].
Lemma 3.1. [9] If U1 , U2 are Σ01 (A) classes and U1 U2 ⊆ V where V is a Σ01 (B)
class of measure less than 1 then at least one of the following holds:
• there exists V1 ∈ Σ01 (B) of measure less than 1 such that U1 ⊆ V1 .
• there exists V2 ∈ Σ01 (B) of measure less than 1 such that U2 ⊆ V2 .
Hence (by iteration), if U is a Σ01 (A) class and U n ⊆ V where V is a Σ01 (B) class
of measure less than 1 then there exists V1 ∈ Σ01 (B) of measure less than 1 such that
U ⊆ V1 .
Proof. It suffices to prove the first clause. First suppose that there is $ such that
&$ (V ) < 1 and $ ∈ U1 . Then define
V2 = {' | $' ∈ V }
and note that U2 ⊆ V2 and &(V2 ) < 1. Otherwise let q be a rational such that
&(V ) < 1 − q, define
V1 = {$ | &$ (V ) > 1 − q}
RANDOMNESS, LOWNESS AND DEGREES
and note that U1 ⊆ V1 and (1 − q) · &(V1 ) ≤ &(V ) (so &(V1 ) < 1).
Theorem 3.2. [9] For all A, B ∈ 2 the following are equivalent:
#
563
-
• A ≤LR B.
• For every Σ01 (A) class T A of measure < 1 there is a Σ01 (B) class V B such that
&(V B ) < 1 and T A ⊆ V B .
• For some member U A of a universal Martin-Löf test relative to A there is
V B ∈ Σ01 (B) such that &(V B ) < 1 and U A ⊆ V B .
Proof. The following proof is essentially the one presented in [9] but without the
use of prefix-free machines. First suppose that A ≤LR B, i.e., that every B-random
is A-random. Let (VeB ) be a universal Martin-Löf test relative to B and let (Ue ) be
the test of Lemma 2.1. Then there must be a string $ and m ∈ N such that [$] (⊆ V0B
and
UmA ∩ [$] ⊆ V0B ∩ [$].
(2)
Indeed, if the negation of this held then we could use it to construct (by finite
extensions) a real which belongs to UeA for all e but does not belong to V0B ; that is,
a B-random real which is not A-random.
Now it is not hard to construct a Σ01 (B) class W B such that &(W B ) < 1 and
A
Um ⊆ W B . We just have to enumerate into W B all reals outside the cone [$] plus
the reals in [$] ∩ V0B . Then by (2) we have UmA ⊆ W B and since &$ (V B ) < 1 we
also have &(W B ) < 1. Now consider a Σ01 (A) class T A of measure < 1 (consider
T A as a prefix-free set of strings). Then some subset of the sequence ((T A )k ) is
a Martin-Löf test relative to A. By the properties of (Ue ) there exists k such that
(T A )k ⊆ UmA . Hence (T A )k ⊆ W B and by Lemma 3.1 this proves the second clause
of the claim.
The second clause obviously implies the third clause so we are left to derive the
first clause from the third clause. Suppose that % is B-random and V A ⊆ T B where
V A is a member of a universal Martin-Löf test relative to A, T B ∈ Σ01 (B) and
&(T B ) < 1. Then the complement of T B contains a tail of %, so the complement
of V A also contains that tail of %. Since V A belongs to a universal test, % is
A-random.
-
The following result shows how two universal oracle Martin-Löf tests are related
(or how ‘similar’ they are) and its proof is in the spirit of the proof of Theorem 3.2.
Theorem 3.3. If (Ui ) is an oracle Martin-Löf test, V is a member of a universal
oracle Martin-Löf test and '0 , $0 ∈ 2<# such that [$0 ] (⊆ ∩!⊃'0 V ! then there exist
', $ ∈ 2<# , m ∈ N such that
• ' ⊃ '0 and $ ⊃ $0 ,
• there is % ⊃ ' such that [$] (⊆ V % ,
• for all ! ⊃ ', Um! ∩ [$] ⊆ V ! .
Proof. Assuming that the claim does not hold we will construct α, * ∈ 2# such
that α ∈ ∩i Ui* and α (∈ V * , which is clearly a contradiction. Let α0 = $0 , *0 = '0
and at stage s > 0 pick strings $ ⊃ αs−1 and ' ⊃ *s−1 such that
• there is ! ∈ ['] such that [$] (⊆ V ! ,
• [$] ⊆ Us'
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G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
and set αs = $, *s = '. These strings will exist by the assumptions about $0 , '0 (for
the first step) and the assumption that the claim does not hold. Now if α = ∪s αs ,
* = ∪s *s then α ∈ ∩i Ui* and α (∈ V * .
A natural question about reducibilities 1 on the reals is to determine the measure
of upper and lower cones. For the Turing reducibility the lower cones are countable
(hence they are null) and the non-trivial upper cones have measure 0 [18]. For ≤LR
although lower cones are not always countable (see Section 4) it is not difficult to
show that they are null.
Theorem 3.4. For every A the set {% | % ≤LR A} has measure 0.
Proof. Given A the A-random numbers have measure 1 and so it is enough to
show that if % is A-random then % (≤LR A. But this is obvious since % is not
%-random.
For the upper cones it is tempting to think that a version of the majority vote
technique which settled the question for ≤T (see [5] for an updated presentation
of the argument) would work for ≤LR (especially if one thinks of randomness in
terms of betting strategies). However Frank Stephan pointed out (in discussions
with the first author) that the answer is most easily given by an application of van
Lambalgen’s theorem (a simple theorem with many applications) which asserts that
A ⊕ B is random iff A is random and B is A-random.
Theorem 3.5 (Frank Stephan). If A is random then it is B-random for almost all
B ∈ 2# . Also, any non-trivial upper cone in the LR degrees has measure 0.
Proof. For the first claim, let A be random. Since the random sets have measure
1, the measure of sets B such that A is B-random equals the measure of random sets
B with the same property. By van Lambalgen’s theorem (and since A is random)
these are the sets B which are A-random. But these have measure 1.
For the second clause it suffices to show that if A is not low for random then the
measure of sets B such that every B-random real is A-random is 0. If A is not low
for random there is a random R which is not A-random. Since (by the first clause)
R is B random for almost all B, the class of B-randoms is not contained in the class
of A-randoms for almost all B.
§4. Uncountable predecessors. In computability theory we are used to structures
in which every degree has only countably many predecessors. Below we show that
the LR degrees do not have this property1 and that, in fact, whenever α is not GL2
the degree of α has an uncountable number of predecessors.
Lemma 4.1. Let U be a member of an oracle Martin-Löf test, n ∈ N and '0 ∈ 2<# .
Then there exists '1 ⊃ '0 such that for all '2 ⊃ '1 , &(U '2 − U '1 ) < 2−n .
Proof. Suppose towards a contradiction that the claim of the lemma does not
hold. Then there exists n ∈ N such that every extension '1 of '0 can be extended
to '2 such that &(U '2 − U '1 ) ≥ 2−n . We can therefore construct by using a finite
number of extensions '1 such that &(U '1 ) > 2−1 .
In [16] (see [20] for a different proof and more detailed presentation) it was shown
that the LR degrees are countable equivalence classes.
1 Joe Miller and Yu Liang have independently announced the existence of an LR degree with uncountably many predecessors.
RANDOMNESS, LOWNESS AND DEGREES
565
Theorem 4.2. In the LR degrees the degree of ∅# bounds 2ℵ0 degrees.
Proof. By cardinal arithmetic it is enough to show that the set B = {% | % ≤LR
∅# } has cardinality 2ℵ0 . Let U be the second member of the universal oracle MartinLöf test of Lemma 2.1, so that by definition &(U % ) < 2−2 for all % ∈ 2# . It suffices
to define a ∅# -computable perfect tree T (as a map from 2<# to itself which preserves
the prefix and incompatibility relations) such that
1
&(A) ≤ where A = ∪'∈T U ' .
2
Then |[T ]| = 2ℵ0 (where [T ] is the set of infinite paths through T ), and for all
% ∈ [T ], U % ⊆ A. Since A is ∅# -c.e., we have by Theorem 3.2 that for all % ∈ [T ],
% ≤LR ∅# . We ask that &(A) ≤ 12 (rather than &(A) < 1) simply in order that figures
used should be in line with what will appear in the proof of Theorem 4.3.
It remains to define such a tree T and verify the construction. First find a string
"
' such that for any extension ' # of ', &(U ' − U ' ) < 2−4 and define T (∅) = '. The
existence of such a string is ensured by Lemma 4.1. Note that &(U T (∅) ) < 2−2 .
Now for each of the one element extensions of T (∅), say 'i , i = 0, 1 find some
"
"
extension 'i# ⊇ 'i such that for any ' # ⊃ 'i# we have &(U ' − U 'i ) < 2−6 . Define
T (0) = '0# , T (1) = '1# and note that &((U T (0) ∪ U T (1) ) − U T (∅) ) < 2 · 2−4 = 2−3
by the previous step. Continue in the same way so that at the n-th stage, where we
define T ($) for all $ with |$| = n, we choose a value ' for T ($) such that for all
"
' # ⊃ ' we have &(U ' − U ' ) < 2−(2n+4) . Let
Cn = {T ($) | $ ∈ 2<# ∧ |$| ≤ n}
and note that Cn ⊆ Cn+1 . Also let
An = ∪'∈Cn U '
and note that An ⊆ An+1 and A = ∪n An . By induction, for all n
n
!
1
&(An ) <
2i · 2−(2i+2) = 2−2 + 2−3 + · · · = .
2
i=0
Note that the factor 2i in the above sum comes from the number of strings of level
i in T (and where we say that ' is of level i in T if ' = T ($) for $ of length i). It
remains to show that we can run the construction of T computably in ∅# , but this
follows immediately from Corollary 2.2.
#
After we proved Theorem 4.2 and since high degrees often resemble 0 , we considered
showing that every high LR degree has uncountably many predecessors. Using a
combination of highness techniques from [8, 13, 19] we succeeded in showing that
if A is generalized superhigh (i.e., A# ≥tt (A ⊕ ∅# )# ) then A has uncountably many
≤LR -predecessors. The following theorem is a stronger result showing that if A
is merely GL2 (i.e., generalized non-low2 , A## >T (A ⊕ ∅# )# ) then it has the same
property. For other GL2 constructions we refer the reader to [13].
Theorem 4.3. If α is GL2 then in the LR degrees the degree of α bounds 2ℵ0
degrees.
Proof. The basic idea behind the proof remains the same as in the proof of
Theorem 4.2 but now we proceed to define T using only an oracle for α rather
than an oracle for ∅# . In order to do so we drop the requirement that T should
566
G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
be perfect—we allow that some strings in T have no extensions in T —and we αapproximate a tree T ∗ ⊆ T which is perfect during the course of the construction.
Once again we let U be the second member of the universal oracle Martin-Löf test
of Lemma 2.1. We ensure that
1
(3)
&(A) ≤ where A = ∪'∈T U ' .
2
The fact that T ∗ is a subtree of T and is perfect ensures that there exist 2ℵ0 infinite
paths through T . To make sure that our approximation to T ∗ converges we will use
the fact that α is GL2 .
The construction will be α-computable and will enumerate a set of strings T such
that (3) holds. Every time we enumerate a string in T we do it in order to extend
T ∗ , wishing that it can be made perfect in the limit. In order to ensure that (3)
held in the proof of Theorem 4.2 we first enumerated one string adding less than
2−2 to &(A), then two strings adding less than 2−4 each to &(A), and then for every
n we enumerated 2n strings adding less than 2−2n−2 each to &(A). Now we extend
that idea. We use movable markers which take positions on strings in the current
version of the tree T . At any stage we shall have at most one marker of the form
l0 , at most two markers of the form l1 , and in general, for any n we have at most
2n markers of the form ln . At every stage of the construction we ensure that the
following condition is satisfied:
For every n there exist at most 2n strings in T allocated a marker ln .
Each string allocated a marker ln adds less than 2−2n−2 to &(A) and
all measure in A can be attributed to some string with a marker (i.e.,
for every ' ∈ T there exists ' # ⊇ ' in T which is allocated a marker).
(4)
Clearly if we ensure that (4) is satisfied at every stage of the construction we shall
have that &(A) ≤ 12 .
Using the fact that α ∈ GL2 . We shall use the fact that for every function f ≤T ∅#
there exists a function h ≤T α which is not dominated by f (i.e., which is larger
than f on an infinite number of arguments). Let the finite binary strings be ordered
by length and then from left to right and let ∅ denote the empty string (at least for
the duration of this proof). Given any ' ∈ 2<# and n ∈ #, let search(', n) be the
least k such that:
• there exists a least '1 ⊇ ' of length < k such that for all '2 ⊃ '1 , &(U '2 −U '1 ) <
2−2n−4 .
• for all '0 ⊇ ' less than this '1 (according to the ordering specified above) there
exists '2 ⊃ '0 of length < k with &(U '2 − U '0 ) ≥ 2−2n−4 .
In other words search(', n) is the length of search required in order to extend any
leaf ' of T correctly at stage n in the proof of Theorem 4.2. Now let f be defined
by recursion as follows:
f(0) = search(∅, 0),
f(n + 1) = max{search(', n + 1) | |'| ≤ f(n)}.
(5)
(6)
The point of the function f is that it bounds the length of search necessary to extend
T “correctly” at stage n of the construction, even if we have not defined T correctly
at previous stages because we did not search for long enough. We take h ≤T α
RANDOMNESS, LOWNESS AND DEGREES
567
which is not dominated by f and at each stage n of the construction we shall now
search for h(n) many steps in order to decide how to extend T (rather than f(n)
many steps, which is in effect what we did before). The fact that h(n) ≥ f(n) for
infinitely many n will suffice to show that our approximation to T ∗ converges.
We are now ready to define the construction—we shall explain how we act in
order to satisfy (4) and how we use the fact that h is not dominated by f to ensure
that our approximation to T ∗ converges as we define the construction.
Stage 0. Let ' be the least string such that for all ' # ⊃ ' of length < h(0),
"
&(U ' − U ' ) < 2−4 . Define T (∅) = T ∗ (∅) = ' and allocate the marker l0 to '.
At stage 0, then, we proceed exactly as we did in the proof of Theorem 4.2 except
that now we only search for h(0) many steps. Clearly (4) is satisfied at the end of
stage 0 since there exists just one string which is allocated a marker l0 and this string
adds less than 2−2 to &(A).
Stage n > 0. By the end of stage n − 1 we shall have decided all values T ($) for
$ of length ≤ n − 1. Since T is partial we may have decided that some of these
values should be undefined. We will also have a present approximation to T ∗ . For
some greatest m ≤ n − 1 we shall have that T ∗ ($) ↓ for all $ of length ≤ m and that
T ∗ ($) ↑ for all $ of length > m. The leaves of T ∗ will be strings of level n − 1 in T .
All strings in T ∗ will be allocated a marker (as will some other strings). Assume
inductively that (4) is satisfied at the end of stage n − 1 and that so far we have only
allocated markers ln" for (some) n # ≤ n − 1.
Step (a). The first thing we do at stage n is to form a set of possible candidates
for strings of level n in T . For each leaf ' of T ∗ we proceed as follows. For each of
the one element extensions of ', say 'i , i = 0, 1 we find the least extension 'i# ⊇ 'i
"
"
such that for any ' # ⊃ 'i# of length < h(n) we have &(U ' − U 'i ) < 2−2n−4 . We let
C be the set of all 'i# chosen in this way (ranging over all leaves ').
Step (b). The second thing we do is to decide which of the strings in C should
be enumerated into T and at which level we should extend T ∗ . As above, let m
be the greatest such that T ∗ ($) ↓ for all $ of length ≤ m. Our first choice is to
extend T ∗ at level m + 1. This is possible if for every leaf ' of T ∗ there exist two
"
distinct extensions ' # ∈ C with &(U ' − U ' ) < 2−2n−2 . In this case we enumerate
all strings in C into T and into T ∗ (so that these are strings of level n in T and
level m + 1 in T ∗ ). Each of these strings is allocated a marker ln . Clearly, in this
case, condition (4) is satisfied since there are at most 2n strings to which we have
just allocated markers ln and each of these strings adds less than 2−2n−2 to &(A).
Otherwise we must extend T ∗ at a level m # ≤ m and we do so at the greatest
level at which extension is possible. For ' ∈ T let p(') be the longest proper initial
segment of ' to which a marker is allocated if there exists such, putting p(') = ∅
otherwise. Extension is possible at level m # if for each ' of level m # in T ∗ there exists
"
an extension, e(') say, in C with &(U e(') − U p(') ) < 2−2n −2 , where n # is such that
' is allocated a marker ln" . If we extend at level m # then for each ' = T ∗ ($) for
$ of length m # we redefine T ∗ ($) = e(') and enumerate e(') into T . We allocate
the marker previously allocated to ' to e(') and remove all markers from strings ' #
with p(') ⊂ ' # ⊂ e('). We make all values T ∗ ($) for $ of length > m # undefined.
Clearly in this case we have still satisfied (4) since any measure in A can be attributed
568
G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
to the same markers as before, except that measure in U e(') − U p(') which can now
"
be attributed to the marker ln" on e(')—which still contributes less than 2−2n −2 .
At the very least we shall always be able to extend T ∗ at level 0. Now suppose
that h(n) ≥ f(n) and we extend T ∗ at level m # at stage n. Then each string ' which
we define as a leaf of T ∗ by the end of stage n really will satisfy the property that
"
for all ' # ⊃ ', &(U ' − U ' ) < 2−2n−4 . Thus at every stage > n we shall be able to
extend T ∗ at a level > m # .
Verification. It follows immediately by induction that at every stage condition (4)
is satisfied, so that the measure of that portion of A already enumerated by the end
of stage n is always less than 12 . Thus &(A) ≤ 12 .
It remains to show that the approximation to T ∗ converges but this follows
immediately from the observation made above, that whenever h(n) ≥ f(n) and we
extend T ∗ at level m # at stage n, we shall be able to extend T ∗ at a level > m # at
every stage > n.
§5. Computably enumerable LR degrees. In this section we study the structure
of the c.e. LR degrees and their relationship with the Turing reducibility. The
results have been chosen so that they demonstrate how to transfer selected basic
techniques from the c.e. Turing degrees (like Sacks coding and restraints) to the c.e.
LR degrees. The reader can use these examples in order to prove new results about
this structure. We note that the relationship between ≤LR and ≤T is nontrivial and
goes beyond what we discuss here. For example there is a half of a minimal pair in
the c.e. Turing degrees which is LR-complete [1, 2]. For background in the theory
of c.e. degrees we refer the reader to [21]. The following theorem demonstrates how
infinitary Sacks coding can be handled in the LR degrees.
Theorem 5.1. If W is an incomplete c.e. set, i.e., ∅# (≤T W , then (uniformly in W )
there is a c.e. set B such that B ≤LR W and B (≤T W .
Proof. A relativisation of the classic non-computable low for random argument
of [12] (also see [5]) merely gives that for all A there exists B c.e. in A such that
B (≤T A and A⊕B ≤LR A. If we assumed that W is low we could prove Theorem 5.1
with a finitary argument similar to [12] by using a lowness technique (namely
Robinson’s trick). To prove the full result we need infinitary coding combined with
cost efficiency considerations (see [14] for examples of cost-function arguments).
We need to construct a c.e. operator V and a c.e. set B such that
UB ⊆ V W
(7)
&(V W ) < 1
(8)
where U is a member of the universal oracle Martin-Löf test of lemma 1 (so that
&(U ) < 2−1 ),
and the following requirements are satisfied
W
#
Pe : ΦW
e = B ⇒ Γe = ∅
where (Φe ) is an effective enumeration of all Turing functionals and Γe are Turing
functionals constructed by us. It is worth being very clear about the timing of
enumerations. We shall suppose that at the beginning of stage s we have already
defined Bs and that enumerations we make during the course of stage s are in order
RANDOMNESS, LOWNESS AND DEGREES
569
to define Bs+1 . At stage s we let bs be defined as follows: let ns be the least number
enumerated into B at stage s − 1 (i.e., the least number in Bs − Bs−1 ) if there exists
such and otherwise let ns = s; then bs = Bs ! ns + 1 and s is a true stage for the
enumeration of B if bs ⊂ B.
The operator V can be defined ahead of the construction. At the beginning of
stage s we enumerate a string $ into V W in the following cases:
• there exists )', $* ∈ U such that ' ⊆ bs , $ is not presently in V W and we
have not previously enumerated $ into V W at any stage s # such that ' ⊆ bs " .
In this case we enumerate $ into V W with big (i.e., larger than any number
mentioned in the construction so far) W -use uW,' ($).
• there exists )', $* ∈ U such that ' ⊆ bs , we have previously enumerated $ into
V W at a stage s # such that ' ⊆ bs " but now $ is not in V W at the beginning of
stage s due to some W -change. In this case we enumerate $ into V W with the
same W -use uW,' ($) as before.
Note that this definition of V ensures that (7) is satisfied. In the following it is helpful
to assume that V W is prefix-free (as a set of finite strings) at all stages. It is clear
that we can arrange the enumeration of V so that this holds, while maintaining
U B ⊆ V W at all stages. The main conflict we face is that on the one hand we
want a Sacks coding for each of the Pe requirements (enumerations into ∅# may
trigger B-enumerations infinitely often) and on the other hand B-enumerations
may force &(V W ) = 1 (via the definition of V given above). The connection
between B-enumerations and superfluous measure in V W (in the sense that it does
not serve (7), it corresponds to intervals which are not in U B ) is roughly as in the
noncomputable low for random construction of [12]: some interval $ is enumerated
into U B with use u, it enters V W with use v and subsequently B ! u changes thus
ejecting $ from U B . Then W ! v could freeze, thus capturing a useless interval in
V W . We already have &(V W ) ≥ &(U B ) so we want to make sure that the measure
corresponding to useless strings is bounded by 2−1 . Here, however, we have an
advantage over the classic argument in [12] as W may also change, thus extracting
the useless string from V W . We will use this fact in order to make infinitary coding
into B possible while satisfying (8). Let
W
+(ΦW
e , B)[s] = max{t | Φe (n)[s] ↓= B(n)[s] for all n ≤ t}
be the length of agreement of the reduction ΦW
e in relation to B at stage s.
Idea behind the Pe -strategy. The functional Γe will be implicit. During the
construction we define codes (markers) kne so that if kne ↓ and n 7 ∅# then we need
to put kne 7 B. Thus ∅# changes are coded into B and B changes are coded into W
W
#
via Φe as in the usual Sacks argument: assuming that ΦW
e = B we get Γe = ∅ , a
e
W
contradiction. Note that kn is defined below the length of agreement +(Φe , B); if
at some stage this length of agreement drops below kne then we are allowed to move
kne . Our commitment is that if the reduction ΦW
e = B is total (i.e., every marker in
this reduction settles down) then every kne reaches a final position.
Strategy Pe can afford cost (i.e., measure of useless strings in V W ) at most
−(e+2)
2
. In this way after the construction we can count the measure of V W as
B
&(U ) plus the junk measure for each Pe which amounts to a number < 1 (for such
cost counting arguments see [14]). The idea behind the strategy is the following:
570
G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
for any x, s we define the absolute cost of x at stage s as
Acost(x, s) = &($ | ∃)', $* ∈ U with ' ⊆ bs , |'| > x and uW,' ($) ↓)
and the relative cost of x at stage s as
Rcoste (x, s) = &($ | ∃)', $* ∈ U with ' ⊆ bs , |'| > x and uW,' ($) ↓< ve (x, s))
where ve (x, s) is the use of ΦW
e (x)[s] if this is defined, and ∞ otherwise. The
absolute cost is the maximum cost we could pay for enumerating x 7 B and it
corresponds to the worst case scenario that there are no helpful W -changes (as
in [12] where W is empty). Note that we would not enumerate a code x into B
unless +(ΦW
e , B) > x. If this diagonalization is unsuccessful then W ! v(x, s)
will change. The relative cost is the expected cost under the assumption that the
Φe -computation will recover.
It is important to note that during our attempts to achieve ΦW
e (= B we will have
to pay the absolute cost at most once but we might have to pay relative costs infinitely
often (due to the infinitary coding). So we split the amount 2−(e+2) allowed into
two and we are going to restrict the absolute cost to 2−(e+3) and the total of relative
costs to 2−(e+3) . Now we split the relative cost allowance to the markers kne that
the strategy operates: the enumeration of kne should not cost more than 2−(e+n+4)
(unless W does not respond to this diagonalization, in which case we can count
the cost of kne as the absolute cost). Notice that kne (whatever values it takes in the
course of the construction) will be enumerated into B at most once (so it will cost
us at most once) since we only enumerate it into B if n 7 ∅# .
If the strategies respect the above quotas when they perform their coding (Benumerations) then (8) is satisfied (by a standard cost-counting argument relative
to W ). Now we only need to show that the Sacks coding can work under the above
cost efficiency restrictions. This will follow from the following lemma, which will
be proved once we have defined the construction.
= B is total then all markers kne such that n ∈
/ ∅# will be
Lemma 5.2. If ΦW
e
permanently defined (despite the tough cost-related conditions).
The proof of this lemma is just a slight extension of the standard cost function
arguments that limx supt cost(x, t) = 0 (see [12, 14]). Based on the assumption
that Lemma 5.2 holds Pe will place the next marker kne onto a number x0 such
that Rcost(x0 , s) is currently less than the relative cost allowance qn for kne . Then
according to the definition of V , in later stages Rcost(x0 ) will remain less than qn
(as new measure in V W comes with big W -use), although the absolute cost may
increase. Thus the coding can go through by keeping the relative cost below quota.
There are some final points that we need to sort out in order to ensure that Pe can
perform the coding under the cost efficiency restrictions. Although we managed to
keep the relative cost below quota, the absolute cost of enumerating some kne ↓ into
B may rise above the threshold 2−(e+3) after we define kne . In fact, ‘false’ measure
in U bs (i.e., strings $ in U bs which are not in the final value U B ) may mean that
this happens for an infinite number of markers at an infinite number of stages of the
construction. The solution is that, once n 7 ∅# , the marker kne will have to wait for
a stage at which the absolute cost drops below quota before being enumerated into
B. Since at every true stage in the enumeration of B we have that U bs is actually
RANDOMNESS, LOWNESS AND DEGREES
571
a subset of the final value U B , for all but a finite number of markers the cost will
drop below quota at all true stages of the construction.
When we define a marker kne we shall choose this value from N[)e,n*] . In order to
ensure that Lemma 5.2 holds we shall allow that when a marker is defined below
the length of agreement we may redefine this value to be less than its previous value.
Clearly this poses no problems in the satisfaction of Pe . When we define a value kne
the value '(kne ) will be the initial segment of Ws below the use of the computation
e
ΦW
e (kn ). Now we state the actual Pe -strategy and the construction. A stage s is
W
called expansionary if +(ΦW
e , B)[s] > +(Φe , B)[t] for all t < s.
Pe -strategy. At stage s let +(ΦW
e , B) be the current length of agreement and do
the following:
1. For each n ∈
/ ∅#s with kne ↓ for which there exists x < kne in N[)e,n*] with
Rcost(x, s) < 2−(e+n+4) redefine kne := x and define '(kne ) to be the initial
segment of Ws below the use in the computation ΦW
e (x).
2. For each kne such that '(kne ) (⊆ Ws make kne ↑.
3. Look for the least n such that n ∈
/ ∅#s , kne ↑ and check if there is x < +(ΦW
e , B),
[)e,n*]
such that Rcost(x, s) < 2−(e+n+4) . If not do nothing, otherwise
x ∈N
define kne = x for the least such x and define '(kne ) to be the initial segment of
Ws below the use in the computation ΦW
e (x). If Pe has not had an expansionary stage subsequent to the last stage at which it made an enumeration then
do nothing more. Otherwise (or if the strategy has not previously enumerated
any markers into B) go to (4).
4. For the least n ∈ ∅#s such that kne ↓∈
/ B and Acost(kne , s) < 2−(e+3) , put kne 7 B
(if there exists such).
Construction. At stage s we access strategies Pi , i < s and let them act through
their steps.
Verification. First we show that (8) is satisfied. Figure 1 illustrates how strings
go in and out of the sets V W [s], U B [s] and V W [s] − U B during the stages s of
the construction. The set V W consists of the strings which have entered V W with
correct W -use. This collection of strings can be divided into two categories: the
ones which came through a B-correct enumeration (as in the definition of the V W enumeration) and the ones which did not. The first category of strings has measure
&(U B ) < 2−1 . The second category strings are the ones which enter the Pe cells
of Figure 1: every time a first category string moves to second category during the
construction it means that this string was in U B and was then extracted due to a
B-change. Such a B-enumeration must have happened in a particular substage by
a particular strategy Pe . Hence this string goes to the Pe cell.
pockets
absolute
cost
... ...
...
P0
...
P1
Figure 1. Junk measure distribution during the construction.
572
G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
Each Pe cell is divided into two main compartments: the first compartment
corresponds to the absolute cost and the second to the relative costs. Every time a
string goes to the Pe cell (as explained above) it first goes to the first compartment
(which may contain many strings but all of them must have come from a single Bchange). Note that this load is associated with a particular marker kne of Pe which
caused the B-change. The second compartment is divided into infinitely many
pockets corresponding to the markers kne , n ∈ N. If and when Pe goes through
a later Φe -expansionary stage, part of the load exits V W (due to the associated
W change) and part of it stays in V W . During such an expansionary stage the
remaining load has measure at most the relative cost of kne and is emptied in the
kne -pocket of the second compartment.
By the cost quotas that we respected in the construction the measure of the
eventual load of the Pe cell will be less than 2−(e+2) . Indeed, the first compartment
of Pe will eventually measure less than the absolute cost quota 2−(e+3)"
and the second
compartment has to measure less than the sum of all relative costs n 2−(e+n+4) =
2−(e+3) . Hence the measure of the strings of the second category is less than
"
−(e+2)
= 2−1 . This shows that &(V W ) < 1.
e2
Next we prove Lemma 5.2. So suppose towards a contradiction that ΦW
e = B
and n is the least such that n ∈
/ ∅# and there is no stage of the construction after
which kne is always defined to take the same value. Let x ∈ N[)e,n*] be such that
&(U B ) − &(U B!x ) < 2−(e+n+4) . Let s0 be large enough such that Bs0 ! x + 1 ⊂ B,
+(ΦW
e , B) > x + 1 at stage s0 and Ws0 ! ve (x, s0 ) ⊂ W . Let s1 > s0 be a true stage in
the enumeration of B. Then the marker kne will be defined to take the same value at
all stages ≥ s1 : if it is already defined it will move to a position ≤ x through step (1);
otherwise it will be newly defined to a value ≤ x; in any case it will be permanently
defined because s1 is a true stage. This gives the required contradiction.
Finally it is now straightforward to argue that all Pe are satisfied. Suppose
that ΦW
= B. Let m be large enough such that for all n ≥ m with n ∈
/ ∅# ,
e
B
B!kne
−(e+3)
e
)<2
(letting kn take its final value). In order to compute
&(U ) − &(U
∅# (n) for any n ≥ m using an oracle for W , run the construction until a stage s is
found such that either n ∈ ∅#s or kne is defined and below the length of agreement
with a W -correct computation. We have that n ∈ ∅# iff it has been enumerated by
stage s.
It is easy to see that the above proof works even if we require U B⊕W ⊆ V W instead
of (7). In that case we obtain B ⊕W ≡LR W , W <T B ⊕W and hence the following
theorem, given that there are T -incomplete sets in the complete LR-degree and the
known embedding results for the c.e. Turing degrees (an antichain is embeddable in
every nontrivial interval).
Theorem 5.3. Every c.e. LR degree contains infinitely many c.e. Turing degrees
(in the form of chains and antichains) and the only T -topped (i.e., containing a
maximum/maximal T -degree) c.e. LR degree is the complete LR degree.
As far as the global structure is concerned, we can get a similar result by relativising
known constructions of low for random degrees. In particular, the relativised
noncomputable low for random construction [12] gives that for every B there is
A which is B-c.e. and A ⊕ B ≡LR B, B <T A ⊕ B; and a slight extension of
RANDOMNESS, LOWNESS AND DEGREES
573
the argument gives that every B is T -below an antichain of T -degrees in the same
LR-degree, hence the following theorem.
Theorem 5.4. Every LR degree contains infinitely many Turing degrees (in the
form of chains and antichains) and no maximal Turing degree.
Next we show a splitting theorem which also shows how Sacks restraints work in
the LR degrees.
Theorem 5.5. If A is c.e. and not low for random then there are c.e. B, C such that
• B ∩ C = ∅,
• B ∪ C = A,
• A (≤LR B and A (≤LR C .
Proof. The main idea is as in the classic Sacks splitting theorem. We just have
to translate the main tools like length of agreement and Sacks restraints to the case
of LR reductions. This will not be a problem as ≤LR is Σ03 . Fix a member U of a
universal oracle Martin-Löf test; an LR reduction is defined via a c.e. operator V
(as opposed to a Turing functional) and a q ∈ Q2 , 0 < q < 1 (Q2 are the rationals
with finite binary expansion) such that &(V % ) < q for all % ∈ 2# (it is easy to see
that there is an effective enumeration of all such pairs V, q). Then A is LR reducible
to B via V, q if
UA ⊆ V B.
(9)
To define the length of agreement +(U A , V B ) of this possible reduction consider
computable enumerations of U, V, A, B. Let ($s ) be a recursive enumeration of all
the finite strings which appear in UtAt at some stage t, and such that each $ appears
once in this list for each time that it appears in UtAt with new use. If $s is enumerated
into this list at stage t, then at stage t # > t we say that $s has remained in U A if it is
A
in Ut " t" with the same use. Now for all s we define +(U A , V B )[s] to be the maximum
n such that the following hold:
• $n [s] ↓ (that is, the nth member of this sequence has been enumerated by
stage s),
• ∀i ≤ n ([$i ] ⊆ VsBs ∨ $i has not remained in U A ).
Assuming the standard hat-trick for the Σ02 approximation of U A (i.e., let ks =
# =
min{x : x ∈ A[s] − A[s − 1]}, or k = s if there are no such x and define A[s]
A
#
A[s] ! k; then (re)define U [s] := {$ : )$, '* ∈ Us for some ' ⊆ A[s]}) it is clear
that reduction (9) is correct iff
lim inf +(U A , V B )[s] = ∞.
s
Now in general, if we wish to destroy a given reduction like (9) where A is a given
c.e. set of nontrivial LR degree and B is enumerated by us, it’s enough if we respect
the following restraint at every stage s:
/ UsAs )].
r(V, q, s) = &t [∀i ≤ +(U A , V B )[s] ([$i ] ⊆ VsBs with B-use < t ∨ $i ∈
Indeed, we claim that if r(V, q, s) is respected for a cofinite set of stages then
lim +(U A , V B )[s] < ∞.
s
(10)
574
G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
To show this suppose after stage s0 the restraint is respected and start enumerating a
set of strings E as follows: at stage s put $i 7 E if i ≤ +(U A , V B )[s] and $i ∈ UsAs .
It is clear that after s0 , +(U A , V B ) does not decrease. Thus, if (10) did not hold,
U A ⊆ E. Also, &(E) < 1 − q by the properties of V and since E ⊆ V B by the
restraints on B. But this contradicts the assumption that A (≤LR ∅, given that E is
computably enumerable.
So there must be some i such that $i is a permanent resident of U A and $i is
never covered by strings in V B . This means that (9) is destroyed and the restraint
comes to a limit. This is all we needed in order to apply the classic Sacks splitting
argument. Consider an effective enumeration (Ve , qe ) of all tuples (V, q) where V is
a c.e. operator (for all % ∈ 2# , V % ⊆ 2<# ) and q ∈ Q2 ∩(0, 1) with &(V % ) < 1−q for
all % ∈ 2# . To prove the theorem we need to enumerate B, C such that A = B ∪ C ,
B ∩ C = ∅ and the following requirements are satisfied:
Re : U A (⊆ VeB ,
Qe : U A (⊆ VeC .
The rest of the proof is as in the Sacks splitting theorem (see [21] for a presentation),
only that the length of agreement and the restraints are defined as above.
§6. Digression: Enumerations of random sets. It is a classical result of computability theory that every level of the arithmetical hierarchy is proper i.e., that
for every n ≥ 1 there exist sets which are Σ0n and which are not ∆0n . Of course it is
also well known (see [6, 17]) that for every n ≥ 1 there exist sets which are n-c.e.
and which are not (n − 1)-c.e. (by 0-c.e. we mean computable). These are called
properly n-c.e. sets. It seems a natural question, therefore, to ask what can be said
along these lines where we restrict ourselves to the case of random sets.
Theorem 6.1. For every m, n ≥ 1 there exist sets which are n-random and which
are properly m-c.e. in ∅(n).
The easiest way to show this is via the following theorem of Joe Miller (currently
available in [14]). Let P be any Π01 class and A ∈ P ; we say that I is an independent
set for A in P if whenever B satisfies the property that for all n ∈
/ I , B(n) = A(n)
we have B ∈ P . Also, a set A is called autoreducible (see [22]) if for each x the
question ‘is x in A?’ can be answered recursively in A, without ever asking the
oracle about x.
Theorem 6.2 (Joe Miller). Suppose that A ∈ P is not autoreducible. Then there
exists an infinite set I which is an independent set for A in P and which is computable
in A# . (Also, no random set is autoreducible.)
To show Theorem 6.1 recall that there exists a Π01 class P consisting only of
random sets. Apply the low basis theorem in order to obtain a random A and also
I which is an independent set for A in P , computable in ∅# . The theorem then
follows simply by carrying out the standard construction of a properly n-c.e. set
within I in place of the identity tree and by relativizing. In the same way, by the
argument that produces properly n-c.e. degrees for every n ≥ 1 (see [3, 7]), one can
see that Theorem 6.1 also holds degree-theoretically: for every m, n ≥ 1 there exist
sets which are n-random, m-c.e. in ∅(n) and Turing equivalent to no (m − 1)-c.e. in
∅(n) set.
RANDOMNESS, LOWNESS AND DEGREES
575
§7. Low for random are ∆02 revisited. These days the fact that low for random sets
are ∆02 is proved via the fact that this class coincides with the K-trivial reals [16] (given
that it is much easier to show that K-trivials are ∆02 , see [5, 14] for a presentation).
However this was originally proved by Nies [15] who showed that for any low for
random real there exists a Π01 class relative to ∅# which contains it and has finitely
many elements. That proof involved prefix-free complexity and in particular, it
constructed a prefix-free machine with certain properties. Below we adapt his
argument to our framework; the main advantage is that we only have to deal with
natural Π01 classes relative to ∅# which stem from the definition of an oracle MartinLöf test. Notice that in the proof of Lemma 2.1 we first constructed an oracle
Martin-Löf test (Ue ) with the property
For every oracle Martin-Löf test (Ve ), uniformly on its c.e. index
(11)
we can compute k ∈ N such that Ve+k ⊆ Ue (as sets of axioms) for
all e.
and then we modified it in order to satisfy the other conditions.
Theorem 7.1. If U is a member of an oracle Martin-Löf test satisfying condition
(11) and T ∈ Σ01 , &(T ) < 1 then there are only finitely many % ∈ 2# with the property
U% ⊆ T.
Proof. It is enough to find some c ∈ N such that in every prefix-free set of strings
of cardinality c there is a string $ such that U $ (⊆ T . We will obtain c as a fixed
point of our construction. Let f be a computable function such that for every c.e.
index e of an oracle Martin-Löf test (Vi ), f(e) is such that Vf(e) ⊆ U . Effectively
in given k we construct a c.e. operator V such that for all % ∈ 2# , &(V % ) < 2−f(k) .
Clearly, given an index for such V (and k) we can effectively get an index of an oracle
Martin-Löf test whose f(k)-th member is V . So by the recursion theorem there is
k such that the corresponding V is the f(k)-th member of the oracle Martin-Löf
test with index k. Hence, V ⊆ U . We build this V such that there are only finitely
many % ∈ 2# with V % ∈ T .
Our strategies α run over all prefix-free collections of strings of cardinality c =
2f(k)+1 . Let q > 0 be a rational such that &(T ) < 1 − q and (nα ) an effective coding
of the strategies into N. Strategy α will use intervals of measure
pα = q · 2−nα · 2−1 .
Construction of V . At stage s consider those strategies α which require attention
and let them act. A strategy α requires attention at stage s if for all ' ∈ α,
V ' [m + 1] ⊆ Ts where m is the last stage that α acted (and 0 otherwise). Also, a
strategy α acts as follows. Ms is a finite set of strings which covers the reals which
are used by stage s. We start with M0 = ∅. The strategy chooses |α| finite sets of
strings Ci ⊆ T from T − Ms−1 which are mutually disjoint (as Σ01 classes) and have
&(Ci ) = pα · 1c . Then it enumerates them into V αi respectively (where αi is the ith
member of α) and into Ms .
Verification. By induction and the choice of p(α) it is easy to see that when a
strategy requires attention it can find the required sets Ci . Indeed, call α pending at
t if it does not require attention at t and let Pt be the set of these strategies. Then
576
G. BARMPALIAS, A. E. M. LEWIS, AND M. SOSKOVA
&(Mt ) is the measure reserved by strategies at t (i.e., the measure enumerated in
∪( Vt( ) and
!
&(Mt ∩ Tt ) <
pα
α∈Pt
since strategy α can only reserve measure pα at a time, and it does not reserve more
clopen sets until the previous ones have appeared in T . Hence
!
&(Tt − Mt ∩ Tt ) > q −
pα
α∈Pt
"
which shows the claim, given that α pα < q. Also, it can be easily shown that if P
is the complement of a finite collection of strings then it can be covered by a finite
collection of strings and so, for any dyadic rational , < &(P) we can effectively find
a finite set C such that &(C ) = , and C ⊆ P.
It is easily seen that for every k the construction corresponding to k gives V such
that for every % ∈ 2# , V % < 2−f(k) , since whenever a strategy α adds measure
"
to V ( for ( ⊂ % it also adds 2f(k) times that measure to V ( for ( # which are
incompatible with (—and it is clear that whenever ( and ( # are incompatible we
"
have V ( ∩ V ( = ∅.
Finally we show that if k is the fixed point mentioned above (so that V ⊆ U ), then
there do not exist c incomparable strings ( such that U ( ⊆ T . Indeed, otherwise
the corresponding strategy α would act at an infinite number of stages putting pα
measure into T each time, a contradiction.
-
Note that if V is a member of an oracle Martin-Löf test and T ∈ Σ01 then the class
{% | V % ⊆ T } is Π01 relative to ∅# , i.e., it consists of the infinite paths through a
0# computable tree. Since the paths through a 0# computable tree with only finitely
many infinite paths are ∆02 by Theorem 3.2 we get Nies’ result [15] that all low for
random sets are ∆02 . However we wish to note that there are members V of universal
oracle Martin-Löf tests for which the property of Theorem 7.1 does not hold, i.e.,
there are infinitely many % with V % ⊆ T .
To construct such a test one first fixes a uniformly computable infinite family
of reals (%i ). We then proceed just as we did in the proof of Lemma 2.1 in order
to define the second element of that test, only that now we avoid enumerating
axioms )', $* such that ' (= ∅, ' ⊂ %i . This is done by replacing this axiom with
axioms )' # , $* where ' # ⊃ ' and ' # (⊆ %i for all i. Fixing the second element of some
universal (non-oracle) test U # we also enumerate all axioms )∅, $* such that $ ∈ U # .
Note that the only reals % such that V % loses reals (in comparison with the original
test) are the ones in the family (%i ). But this is no harm as for computable %, the
%-random reals are the ∅-random reals. However by Theorem 3.2 and the fact that
the class of low for random reals is countable, if V is a member of a universal oracle
Martin-Löf test there can only be countably many % such that V % ⊆ T .
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UNIVERSITY OF LEEDS
SCHOOL OF MATHEMATICS
LEEDS, LS2 9JT, UK
E-mail: [email protected]
UNIVERSITA DEGLI STUDI DI SIENA
DIPARTIMENTO DI SCIENZE MATEMATICHE ED INFORMATICHE
VIA DEL CAPITANO 15, 53100 SIENA, ITALY
E-mail: [email protected]
UNIVERSITY OF LEEDS
SCHOOL OF MATHEMATICS
LEEDS, LS2 9JT, UK
E-mail: [email protected]